Quasistationarity in a long-range interacting model of particles moving on a sphere

Abstract

We consider a long-range interacting system of $N$ particles moving on a spherical surface under an attractive Heisenberg-like interaction of infinite range, and evolving under deterministic Hamilton dynamics. The system may also be viewed as one of globally coupled Heisenberg spins. In equilibrium, the system has a continuous phase transition from a low-energy magnetized phase, in which the particles are clustered on the spherical surface, to a high-energy homogeneous phase. The dynamical behavior of the model is studied analytically by analyzing the Vlasov equation for the evolution of the single-particle distribution, and numerically by direct simulations. The model is found to exhibit long lived non-magnetized quasistationary states (QSSs) which in the thermodynamic limit are dynamically stable within an energy range where the equilibrium state is magnetized. For finite $N$, these states relax to equilibrium over a time that increases algebraically with $N$. In the dynamically unstable regime, non-magnetized states relax fast to equilibrium over a time that scales as $\log N$. These features are retained in presence of a global anisotropy in the magnetization.

9 pages, 4 figures; v2: refs. added, published version

References in corpus (9)

• Statistical mechanics and dynamics of solvable models with long-range interactions
• A maximum entropy principle explains quasi-stationary states in systems with long-range interactions: the example of the Hamiltonian Mean Field model
• Long time behavior of quasi-stationary states of the Hamiltonian Mean-Field model
• Hamiltonian dynamics reveals the existence of quasi-stationary states for long-range systems in contact with a reservoir
• Incomplete equilibrium in long-range interacting systems
• Relaxation to thermal equilibrium in the self-gravitating sheet model
• Relaxation times of unstable states in systems with long range interactions
• Microcanonical quasi-stationarity of long-range interacting systems in contact with a heat bath
• Action diffusion and lifetimes of quasistationary states in the Hamiltonian Mean Field model

Cited by in corpus (5)

• The world of long-range interactions: A bird's eye view
• Controlling the Range of Interactions in the Classical Inertial Ferromagnetic Heisenberg Model: Analysis of Metastable States
• Long-range interactions, doubling measures and Tsallis entropy
• Classical Heisenberg spins with long-range interactions: Relaxation to equilibrium for finite systems
• Mass Segregation Phenomena using the Hamiltonian Mean Field Model